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Similarity of figures is a property that is used to compare figures. Two triangles are said to be similar if their corresponding sides are in equal ratio and if their corresponding angles are equal. If the corresponding sides of two triangles are equal, then these triangles are called equiangular triangles. In an equiangular triangle, the ratio of the corresponding sides of the two sides is always the same. This property of triangles follows from the basic proportionality theorem.
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These NCERT solutions of Class 10 maths ex 6.2 contain two theorems: (Theorem 1) When a parallel line is drawn along one of the triangle's sides and intersects the other two sides at certain points, the two sides are divided in the same proportion. (Theorem 2) Any line that divides the two sides of a triangle in the same ratio is parallel to the third side. NCERT solutions for exercise 6.2 Class 10 Maths chapter 6 Triangles discuss the concept as per the NCERT Book, like a triangle as a polygon, the interior angles of a triangle, the exterior angles of a triangle, Pythagoras' theorem, and conditions of similarity in triangles.
Triangles Class 10 Chapter 6 Exercise: 6.2
Q1 In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
Answer:
(i)
Let EC be x
Given: DE || BC
By using the proportionality theorem, we get
$\frac{AD}{DB}=\frac{AE}{EC}$
$\Rightarrow \frac{1.5}{3}=\frac{1}{x}$
$\Rightarrow x=\frac{3}{1.5}=2\, cm$
$\therefore EC=2\, cm$
(ii)
Let AD be x
Given: DE || BC
By using the proportionality theorem, we get
$\frac{AD}{DB}=\frac{AE}{EC}$
$\Rightarrow \frac{x}{7.2}=\frac{1.8}{5.4}$
$\Rightarrow x=\frac{7.2}{3}=2.4\, cm$
$\therefore AD=2.4\, cm$
Answer:
(i)
Given :
PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
$\frac{PE}{EQ}=\frac{3.9}{3}=1.3\, cm$ and $\frac{PF}{FR}=\frac{3.6}{2.4}=1.5\, cm$
We have
$\frac{PE}{EQ} \neq \frac{PF}{FR}$
Hence, EF is not parallel to QR.
Answer:
(ii)
Given :
PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
$\frac{PE}{EQ}=\frac{4}{4.5}=\frac{8}{9}\, cm$ and $\frac{PF}{FR}=\frac{8}{9}\, cm$
We have
$\frac{PE}{EQ} = \frac{PF}{FR}$
Hence, EF is parallel to QR.
Q2 (3) E and F are points on the sides PQ and PR respectively of a triangle PQR. For each of the following cases, state whether EF || QR : PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
Answer:
(iii)
Given:
PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
$\frac{PE}{PQ}=\frac{0.18}{1.28}=\frac{9}{64}\, cm$ and $\frac{PF}{PR}=\frac{0.36}{2.56}=\frac{9}{64}\, cm$
We have
$\frac{PE}{EQ} = \frac{PF}{FR}$
Hence, EF is parallel to QR.
Q3 In Fig. 6.18, if LM || CB and LN || CD, prove that $\frac{AM}{AB} = \frac{AN}{AD }$
Answer:
Given : LM || CB and LN || CD
To prove :
$\frac{AM}{AB} = \frac{AN}{AD }$
Since , LM || CB so we have
$\frac{AM}{AB}=\frac{AL}{AC}............(1)$
Also, LN || CD
$\frac{AL}{AC}=\frac{AN}{AD}..............(2)$
From equations 1 and 2, we have
$\frac{AM}{AB} = \frac{AN}{AD }$
Hence proved.
Q4 In Fig. 6.19, DE || AC and DF || AE. Prove that BF / FE = BE / EC
Answer:
Given: DE || AC and DF || AE.
To prove:
$\frac{BF}{FE} = \frac{BE}{EC }$
Since , DE || AC so we have
$\frac{BD}{DA}=\frac{BE}{EC}...........(1)$
Also, DF || AE
$\frac{BD}{DA}=\frac{BF}{FE}.............(2)$
From equations (1) and (2), we have
$\frac{BF}{FE} = \frac{BE}{EC }$
Hence proved.
Q5 In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.
Answer:
Given: DE || OQ and DF || OR.
To prove EF || QR.
Since DE || OQ, we have
$\frac{PE}{EQ}=\frac{PD}{DO}..........(1)$
Also, DF || OR
$\frac{PF}{FR}=\frac{PD}{DO}.............(2)$
From equations (1) and (2), we have
$\frac{PE}{EQ} = \frac{PF}{FR }$
Thus, EF || QR. (Converse of the basic proportionality theorem)
Hence proved.
Answer:
Given: AB || PQ and AC || PR
To prove: BC || QR
Since, AB || PQ so we have
$\frac{OA}{AP}=\frac{OB}{BQ}............(1)$
Also, AC || PR
$\frac{OA}{AP}=\frac{OC}{CR}..............(2)$
From equations (1) and (2), we have
$\frac{OB}{BQ} = \frac{OC}{CR }$
Therefore, BC || QR. (converse basic proportionality theorem)
Hence proved.
Answer:
Let PQ be a line passing through the midpoint of line AB and parallel to line BC, intersecting line AC at point Q.
i.e. $PQ||BC$ and $AP=PB$ .
Using the basic proportionality theorem, we have
$\frac{AP}{PB}=\frac{AQ}{QC}.........(1)$
Since $AP=PB$
$\frac{AQ}{QC}=\frac{1}{1}$
$\Rightarrow AQ=QC$
$\therefore$ Q is the midpoint of AC.
Answer:
Let P is the midpoint of line AB, and Q is the midpoint of line AC.
PQ is the line joining midpoints P and Q of lines AB and AC, respectively.
i.e. $AQ=QC$ and $AP=PB$ .
We have,
$\frac{AP}{PB}=\frac{1}{1}..........................1$
$\frac{AQ}{QC}=\frac{1}{1}...................................2$
From equations (1) and (2), we get
$\frac{AQ}{QC}=\frac{AP}{PB}$
Therefore, by the basic proportionality theorem, we have $PQ||BC$
Answer:
Draw a line EF passing through point O such that $EO||CD\, \, and\, \, FO||CD$
To prove:
$\frac{AO}{BO} = \frac{CO}{DO}$
In $\triangle ADC$ , we have $CD||EO$
So, by using the basic proportionality theorem,
$\frac{AE}{ED}=\frac{AO}{OC}........................................1$
In $\triangle ABD$ , we have $AB||EO$
So, by using the basic proportionality theorem,
$\frac{DE}{EA}=\frac{OD}{BO}........................................2$
Using equations (1) and (2), we get
$\frac{AO}{OC}=\frac{BO}{OD}$
$\Rightarrow \frac{AO}{BO} = \frac{CO}{DO}$
Hence proved.
Answer:
Draw a line EF passing through point O such that $EO||AB$
Given :
$\frac{AO}{BO} = \frac{CO}{DO}$
In $\triangle ABD$ , we have $AB||EO$
So, by using the basic proportionality theorem,
$\frac{AE}{ED}=\frac{BO}{DO}............(1)$
However, it is given that
$\frac{AO}{CO} = \frac{BO}{DO}.............(2)$
Using equations (1) and (2), we get
$\frac{AE}{ED}=\frac{AO}{CO}$
$\Rightarrow EO||CD$ (By basic proportionality theorem)
$\Rightarrow AB||EO||CD$
$\Rightarrow AB||CD$
Therefore, ABCD is a trapezium.
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1. Similar Figures: Two figures are said to be similar if their corresponding sides are in the same proportion and corresponding angles are equal.
2. Conditions for Similar Triangles: The criteria for similarity of the triangle are AA (Angle-Angle), in which two angles of one triangle are equal to the corresponding two angles of another triangle.
3. Midpoint Theorem: In this theorem, the line segment that joins the midpoint of the two sides of the triangle is parallel to the third side of the triangle, and the length of this line segment is half of the third side of the triangle.
4. Parallel Lines and Transversals: A transversal divides the sides of the triangle into proportional segments when it intersects the two parallel lines.
Also see-
Students must check the NCERT solutions for Class 10 Maths and Science given below:
Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:
Frequently Asked Questions (FAQs)
The ratio of the areas of the triangles is equal to the square of the ratio of the sides.
Ratio of area = 1:4
4/x=1/3
x=12cm
The length of the longer side is 12cm.
It will always be parallel to the third side of the triangle.
The congruent figures are all similar, but the opposite is not true.
If we know the side ratio and the length of any side, we can find the length of the corresponding side of the other triangle by using the ratio.
The ratio of the areas of the triangles is equal to the square of the ratio of the sides.
No, resemblance only relates to the shape of the figures and does not imply equality.
The term "similar" refers to two figures that have the same shape but differ in size.
On Question asked by student community
Hello,
Yes, you can give the CBSE board exam in 2027.
If your date of birth is 25.05.2013, then in 2027 you will be around 14 years old, which is the right age for Class 10 as per CBSE rules. So, there is no problem.
Hope it helps !
Hello! If you selected “None” while creating your APAAR ID and forgot to mention CBSE as your institution, it may cause issues later when linking your academic records or applying for exams and scholarships that require school details. It’s important that your APAAR ID correctly reflects your institution to avoid verification problems. You should log in to the portal and update your profile to select CBSE as your school. If the system doesn’t allow editing, contact your school’s administration or the APAAR support team immediately so they can correct it for you.
Hello Aspirant,
Here's how you can find it:
School ID Card: Your registration number is often printed on your school ID card.
Admit Card (Hall Ticket): If you've received your board exam admit card, the registration number will be prominently displayed on it. This is the most reliable place to find it for board exams.
School Records/Office: The easiest and most reliable way is to contact your school office or your class teacher. They have access to all your official records and can provide you with your registration number.
Previous Mark Sheets/Certificates: If you have any previous official documents from your school or board (like a Class 9 report card that might have a student ID or registration number that carries over), you can check those.
Your school is the best place to get this information.
Hello,
It appears you are asking if you can fill out a form after passing your 10th grade examination in the 2024-2025 academic session.
The answer depends on what form you are referring to. Some forms might be for courses or examinations where passing 10th grade is a prerequisite or an eligibility criteria, such as applying for further education or specific entrance exams. Other forms might be related to other purposes, like applying for a job, which may also have age and educational requirements.
For example, if you are looking to apply for JEE Main 2025 (a competitive exam in India), having passed class 12 or appearing for it in 2025 are mentioned as eligibility criteria.
Let me know if you need imformation about any exam eligibility criteria.
good wishes for your future!!
Hello Aspirant,
"Real papers" for CBSE board exams are the previous year's question papers . You can find these, along with sample papers and their marking schemes , on the official CBSE Academic website (cbseacademic.nic.in).
For notes , refer to NCERT textbooks as they are the primary source for CBSE exams. Many educational websites also provide chapter-wise revision notes and study material that align with the NCERT syllabus. Focus on practicing previous papers and understanding concepts thoroughly.
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